IndisputableMonolith.Foundation.SimplicialLedger.ContinuumTheorem

   1import Mathlib.Data.Real.Basic
   2import Mathlib.Analysis.SpecialFunctions.Exp
   3import Mathlib.Analysis.SpecialFunctions.Log.Basic
   4import IndisputableMonolith.Constants
   5import IndisputableMonolith.Foundation.SimplicialLedger.ContinuumBridge
   6import IndisputableMonolith.Foundation.SimplicialLedger.EdgeLengthFromPsi
   7import IndisputableMonolith.Foundation.SimplicialLedger.CubicDeficitDischarge
   8
   9/-!
  10# Continuum Theorem: Field-Curvature Identity at κ = κ_Einstein
  11
  12This module is Phase D of the program to make the paper's Theorem 5.1
  13(field-curvature identity) an unconditional Lean theorem. It composes:
  14
  15- Phase A (`CubicDeficitDischarge.cubic_linearization_discharge`): the
  16  exact discharge of `ReggeDeficitLinearizationHypothesis` on the cubic
  17  lattice.
  18- Phase B (`EdgeLengthFromPsi.jcost_to_regge_factor_eq_kappa_einstein`):
  19  the identification of the bridge normalization with the Einstein
  20  coupling `κ_Einstein = 8 φ⁵`.
  21- The pre-existing upstream quadratic-remainder bound
  22  `J_log_quadratic_approx` (proved in `IndisputableMonolith/Gravity/
  23  CubicReggeProof.lean` and replayed through `IndisputableMonolith`).
  24
  25## What this adds
  26
  27Two statements, both unconditional:
  28
  291. **The exact linearized identity with the Einstein coupling**. The
  30   J-cost Dirichlet energy `laplacian_action G ε` equals `(1/κ_Einstein) ·
  31   (Regge sum)` where the Regge sum is on the conformal edge-length field
  32   generated by `ε`. This is the paper's Theorem 5.1 with the coupling
  33   constant identified (not just `jcost_to_regge_factor`, but the actual
  34   Einstein coupling appearing in `G_{μν} + Λ g_{μν} = κ T_{μν}`).
  35
  362. **The unified certificate** `ContinuumFieldCurvatureCert` bundling:
  37   the discharge, the identity, the coupling value, positivity, and the
  38   flat-vacuum baseline. This is the single artifact downstream papers
  39   can cite when invoking the field-curvature identity.
  40
  41Zero `sorry`, zero new `axiom`.
  42
  43## Scope
  44
  45Phase D is *not* a quantitative `O(a²)` convergence statement for the
  46full nonlinear Regge action on a refined triangulation; that is the
  47subject of Phase C (Cayley-Menger + Schläfli + Piran-Williams) combined
  48with the classical Cheeger-Müller-Schrader (1984) theorem, already
  49axiomatized in `NonlinearConvergence.lean`.
  50
  51What Phase D *does* is close the coupling-identification loop: the
  52bridge theorem's κ is the same κ that appears in Einstein's equations,
  53which is the content of the draft paper's Theorem 6.1 ("Derived Coupling
  54Constant"). Once this is stated with `κ_Einstein` rather than
  55`jcost_to_regge_factor`, the paper's §6 claim "κ = 8 φ⁵ is derived, not
  56fitted" is a Lean theorem composing `kappa_einstein_eq` (already a
  57theorem in `Constants.lean`) with the bridge identity.
  58-/
  59
  60namespace IndisputableMonolith
  61namespace Foundation
  62namespace SimplicialLedger
  63namespace ContinuumTheorem
  64
  65open Constants Cost ContinuumBridge EdgeLengthFromPsi CubicDeficitDischarge
  66
  67noncomputable section
  68
  69/-! ## §1. The field-curvature identity with κ = κ_Einstein -/
  70
  71/-- **THE FIELD-CURVATURE IDENTITY (Einstein-coupling form).**
  72
  73    For any weighted ledger graph `G`, any conformal spacing `a > 0`,
  74    and any log-potential field `ε`:
  75
  76    `laplacian_action G ε = (1 / κ_Einstein) · regge_sum ...`
  77
  78    where κ_Einstein is the Einstein gravitational coupling `8πG/c⁴ = 8 φ⁵`
  79    in RS-native units. This is the paper's Theorem 5.1 combined with
  80    Theorem 6.1: the bridge normalization equals the physical Einstein
  81    coupling. -/
  82theorem field_curvature_identity_einstein {n : ℕ} (a : ℝ) (ha : 0 < a)
  83    (G : WeightedLedgerGraph n) (ε : LogPotential n) :
  84    laplacian_action G ε
  85    = (1 / Constants.kappa_einstein) *
  86        regge_sum (cubicDeficitFunctional n)
  87          (conformal_edge_length_field a ha ε) (cubicHinges G) := by
  88  rw [← jcost_to_regge_factor_eq_kappa_einstein]
  89  exact field_curvature_identity_cubic a ha G ε
  90
  91/-- **COROLLARY.** The flat vacuum `ε ≡ 0` has zero J-cost Dirichlet
  92    energy, and therefore (by the identity) the Regge sum on the
  93    conformal edge-length field of the flat vacuum also vanishes. -/
  94theorem flat_regge_sum_zero {n : ℕ} (a : ℝ) (ha : 0 < a)
  95    (G : WeightedLedgerGraph n) :
  96    regge_sum (cubicDeficitFunctional n)
  97      (conformal_edge_length_field a ha (fun _ => (0 : ℝ))) (cubicHinges G)
  98    = 0 := by
  99  have h_disch : regge_sum (cubicDeficitFunctional n)
 100      (conformal_edge_length_field a ha (fun _ => (0 : ℝ))) (cubicHinges G)
 101      = jcost_to_regge_factor * laplacian_action G (fun _ => (0 : ℝ)) :=
 102    cubic_linearization_discharge a ha G (fun _ => (0 : ℝ))
 103  rw [h_disch, laplacian_action_flat, mul_zero]
 104
 105/-! ## §2. The composed bridge chain
 106
 107For documentation: this is the complete assembly of the cubic-lattice
 108bridge from the Recognition Composition Law to Einstein's field equations
 109in the linearized regime. -/
 110
 111/-- **THE COMPLETE CUBIC BRIDGE CHAIN.**
 112
 113    Given the RCL → J uniqueness → φ forced → constants (all proved
 114    upstream), the linearization identity holds exactly with the Einstein
 115    coupling and vanishes at the flat vacuum. -/
 116theorem bridge_chain_complete {n : ℕ} (a : ℝ) (ha : 0 < a)
 117    (G : WeightedLedgerGraph n) :
 118    -- Discharge: `ReggeDeficitLinearizationHypothesis` holds.
 119    ReggeDeficitLinearizationHypothesis
 120      (cubicDeficitFunctional n) a ha (cubicHinges G) G ∧
 121    -- Identity: J-cost Dirichlet energy = (1/κ_E) · Regge sum.
 122    (∀ ε : LogPotential n,
 123      laplacian_action G ε
 124      = (1 / Constants.kappa_einstein) *
 125          regge_sum (cubicDeficitFunctional n)
 126            (conformal_edge_length_field a ha ε) (cubicHinges G)) ∧
 127    -- Flat vacuum: both sides zero.
 128    (laplacian_action G (fun _ => (0 : ℝ)) = 0 ∧
 129      regge_sum (cubicDeficitFunctional n)
 130        (conformal_edge_length_field a ha (fun _ => (0 : ℝ))) (cubicHinges G) = 0) ∧
 131    -- Coupling value: κ_Einstein = 8 φ⁵.
 132    Constants.kappa_einstein = 8 * Constants.phi ^ (5 : ℝ) ∧
 133    -- Coupling positivity: κ_Einstein > 0.
 134    0 < Constants.kappa_einstein := by
 135  refine ⟨cubic_linearization_discharge a ha G, ?_, ?_, Constants.kappa_einstein_eq,
 136          Constants.kappa_einstein_pos⟩
 137  · intro ε
 138    exact field_curvature_identity_einstein a ha G ε
 139  · refine ⟨laplacian_action_flat G, flat_regge_sum_zero a ha G⟩
 140
 141/-! ## §3. Master certificate -/
 142
 143/-- **CONTINUUM FIELD-CURVATURE CERTIFICATE.**
 144
 145    The single artifact to cite when invoking the field-curvature
 146    identity with the Einstein coupling. Combines:
 147
 148    - discharge of the linearization hypothesis,
 149    - exact identity `∑ψ/∑κ = 1/κ_Einstein · ∑regge`,
 150    - flat-vacuum consistency,
 151    - coupling value κ_Einstein = 8 φ⁵,
 152    - coupling positivity.
 153-/
 154structure ContinuumFieldCurvatureCert where
 155  discharge : ∀ {n : ℕ} (a : ℝ) (ha : 0 < a) (G : WeightedLedgerGraph n),
 156    ReggeDeficitLinearizationHypothesis
 157      (cubicDeficitFunctional n) a ha (cubicHinges G) G
 158  identity : ∀ {n : ℕ} (a : ℝ) (ha : 0 < a) (G : WeightedLedgerGraph n)
 159    (ε : LogPotential n),
 160    laplacian_action G ε
 161    = (1 / Constants.kappa_einstein) *
 162        regge_sum (cubicDeficitFunctional n)
 163          (conformal_edge_length_field a ha ε) (cubicHinges G)
 164  flat_jcost : ∀ {n : ℕ} (G : WeightedLedgerGraph n),
 165    laplacian_action G (fun _ => (0 : ℝ)) = 0
 166  flat_regge : ∀ {n : ℕ} (a : ℝ) (ha : 0 < a) (G : WeightedLedgerGraph n),
 167    regge_sum (cubicDeficitFunctional n)
 168      (conformal_edge_length_field a ha (fun _ => (0 : ℝ))) (cubicHinges G) = 0
 169  kappa_value : Constants.kappa_einstein = 8 * Constants.phi ^ (5 : ℝ)
 170  kappa_pos : 0 < Constants.kappa_einstein
 171
 172theorem continuumFieldCurvatureCert : ContinuumFieldCurvatureCert where
 173  discharge := fun a ha G => cubic_linearization_discharge a ha G
 174  identity := fun a ha G ε => field_curvature_identity_einstein a ha G ε
 175  flat_jcost := fun G => laplacian_action_flat G
 176  flat_regge := fun a ha G => flat_regge_sum_zero a ha G
 177  kappa_value := Constants.kappa_einstein_eq
 178  kappa_pos := Constants.kappa_einstein_pos
 179
 180end
 181
 182end ContinuumTheorem
 183end SimplicialLedger
 184end Foundation
 185end IndisputableMonolith
 186